Characteristics of a Polynomial Function

Did you know???

- Negative leading coefficients result in a negative finite difference. Similarly, a positive leading coefficient results in a positive finite difference.

   i.e. y = -2x³ Since -2 is the leading coefficient and is a negative number, hence, the constant finite 

         difference for this polynomial function will also be a negative constant

- The number of finite differences is the same as the degree of the polynomial function.

   i.e. y = -2x³ Since the degree of the polynomial function is 3, hence, this polynomial function has 3

         finite difference / hence, this polynomial function reaches its constant at the 3rd finite difference

- Simple??? I say very this is pretty straight forward. What makes it simpler is the existence of a formula for finding finite differences without using a graphing calculator!

                                      Constant Finite Difference = Leading Coefficient x Degree!

C.F.D = a x n!

For example,

A polynomial function has 4 finite difference and its constant is -48

Therefore, n = 4   >>>   Remember? The number of finite differences = The degree of a P. function

Using the formula,

C.F.D = a x n!

    -48 = a x (4 x 3 x 2 x 1)

       a = -2

Don't believe me? Try more questions Here

At first, I didn't quite understand how an even/odd degree polynomial function affected the maximum/minimum points of its graph. But after i read through this table, everything seemed to make MUCH more sense! Hoorayyy!!!

Description	Odd Degree (n)		Even Degree (2n)
Leading Coefficient	+	-	+	-
General Shape	Refer to picture A	A reflection in the x-axis of its original shape	A reflection in the x-axis of its original shape	Refer to picture B
Domain	{ x ϵ R }	{ x ϵ R }	{x ϵ R }	{ x ϵ R }
Range	{ y ϵ R }	{ y ϵ R }	{ y ϵ R | y has a restriction }	{ y ϵ R | y has a restriction }
End Behavior	Extends from quadrant 3 to quadrant 1	Extends from quadrant 2 to quadrant 4	Extends from quadrant 2 to quadrant 1	Extends from quadrant 3 to quadrant 4
Number of absolute maximum / minimum points	0	0	1	1
Total number of local maximum / minimum points	Only a maximum of n-1	Only a maximum of n-1	Only a maximum of n-1	Only a maximum of n-1
Total number of x-intercepts	Maximum n, minimum 1	Maximum n, minimum 1	Maximum n, minimum 0	Maximum n, minimum 0

Note :

1) Both ends of the graph with an odd degree faces an opposite direction (1 north, 1 south). Whereas, the ends of the graph with an even degree faces the same direction (both north / both south).

2) ONLY the even degree has :

   - restrictions for its range (y values)

   - ONE absolute maximum / minimum point (ABSOLUTE basically means MOST. So, absolute

     maximum/ minimum point is the point on the graph with even degree that has the highest or lowest

     peak)

3) ALL the types of polynomial graphs can have a maximum number of local maximum / minimum

    points of n-1. Nothing more than that!!!

*** By the way, an absolute maximum / minimum point need not be a turning point on a graph. It can also be an end of the graph which has not restrictions***

I hope this piece of information helped you well! It cleared my doubts, totally!

See you soon!

Power Functions



POWER functions



Power functions are basically the simplest of all polynomial functions. Very basic, but can be tricky at times. Just read the functions carefully, and be sure not to overlook any detail because every part of a power function is significant.

It's a lot easier than you think to imagine the graph of a power function just by STARING at a power function. Of courseeeeeeeeeeeeee, you need some basic knowledge to be able to do magic! No worries, the table below explains everything.

 

 

Key Features	y = xn ; where n is an odd number	y = xn ; where n is an even number
Domain	{ x ϵ R }	{ x ϵ R }
Range	{ y ϵ R }	{ y ϵ R | y ≥ 0 }
Symmetry	Point symmetry about (0,0)	Line symmetry at x = 0
End Behavior	As x → ∞ , y → ∞ As x → -∞ , y → -∞	As x → ∞ , y → ∞ As x → -∞ , y → ∞

Before further explanation, there are new terms that need to be remembered as it will help in understanding future polynomial questions.

          Degree = The highest power in a power / polynomial function

               e.g. y = 3x² + 2x + 1    this means, the degree of the given power function is 2 (since 2 > 1)

          Leading coefficient = The coefficient of the term with the highest power in a power/polynomial function

               e.g. y = 3x² + 2x + 1    this means, the leading coefficient is 3, since it is the coefficient of x²

Tip :

1) The range of a power function remains +∞ at all times ONLY for a power function with an even degree.

2) Only the range of a power function with an even degree has restrictions.

3) For a point symmetry, the coordinate of that point must be stated to show a clearer understanding of the question. Whereas for a line symmetry, the value of x at the symmetry must be stated. (note: x need not necessarily be 0. It can be a whole number depending on the power function)

To practise, try imagining the shapes of the graphs just by analyzing the power function given. And when you've become an expert, try vice versa. When you're an expert in both the ways, you deserve a pat on the shoulder! =)

Wait, wait, wait. It's not the end yet! Just a liiiiittle more...

End Behavior	Function
Extends from quadrant 3 to quadrant 1	y = xn where n is an odd number (if the coefficient of x is a positive)
Extends from quadrant 2 to quadrant 4	y = -xn where the coefficient of x is a negative number (if n is an odd number)
Extends from quadrant 2 to quadrant 1	y = xn where n is an even number (if the coefficient of x is a positive)
Extends from quadrant 3 to quadrant 4	y = -xn where the coefficient of x is a negative number (if n is an even number)

Note that polynomial graphs vary by THEIR DEGREE (odd/even) & THE SIGN OF THEIR LEADING COEFFICIENT (positive/negative)

If you intend to use a graphing calculator, refer to this link Click Here for a step-by-step guideline on how you can plot a graph by just keying in a function in your graphing calculator (I am referring to Texas Instrument TI-84 which looks like below)

 

Happy Math! Until next time...